Structural Analysis of Controlled Switching Linear Hybrid Systems Modelled by Bond Graph

Volume 3, Issue 3, September 2013



Abstract— Switching systems are very common in various engineering fields (e.g. hydraulic systems with valves,.., electric systems with diodes, relays,…, mechanical systems with clutches...). Such systems are a particular case of hybrid systems.

These systems are characterized by a Finite State Automaton (FSA) and a set of dynamic systems, each one corresponding to a state of the FSA. The change of states can be either controlled or autonomous. The aim of this work is to investigate the structural controllability for controlled switching linear hybrid systems modelled by bond graph.

Several concepts appeared in the last decade addressing the controllability problem of these systems: controllable sublanguage concept [9], hybrid controllability concept [10], between-block controllability concept [11]. Controlled switching linear hybrid systems (CSLHS) on which we focus in this work belong to the hybrid controllability concept as they address a reachability problem of hybrid states.

In the other hand, the bond graph concept is an alternate representation of physical systems. Some recent works permit to highlight structural properties. In [7], the structural controllability property is studied using simple causal manipulations on the bond graph model. The objective of this work is to extend these properties to CSLHS systems. The bond graph structure junction contains information’s on the type of the elements constituting the system, and how they are interconnected, whatever the numerical values of parameters.

The structural controllability of CSLHS is studied using simple causal manipulations on the bond graph model. For that, formal representation of structural controllability subspace, is given for bond graph model. It is calculated using causal manipulations.

The base of this subspace is used to propose a graphical procedure to study the structural controllability.

Index Terms— Bond Graph Approach, Hybrid Systems, Controlled Switching Linear hybrid Systems.

I. INTRODUCTION

As a particular class of hybrid control systems, a controlled switching linear hybrid systems consists of several linear subsystems and a rule that orchestrates the switching among them. Switching between different subsystems or different controllers can greatly enrich the control strategies and may achieve better control performances than fixed controllers.

Generally, these systems are characterized by a FSA and a set of dynamic systems, each one corresponding to a state of the FSA. The change of states can be either controlled or autonomous. Various researchers investigated this problem

using the bond graph tool [1]-[2]-[3]-[4]-[5]-[6]. The ideal and the non-ideal approaches are used :

- In the non-ideal approach, switches are modelled as resistive elements associated with modulated transformer. The modulation is done using a Boolean variable.

- In the ideal approach, switches commutate instantaneously. Each switch is modelled as a null source: effort source for a closed switch state, and flow source for an open one. This approach is used in this work.

Besides, much work has been done on the controllability of controlled switching hybrid linear systems. see e.g., [13]-[16]-[17]-[18]-[19]. The aim of this work is to investigate the structural controllability for controlled switching linear systems modelled by bond graph.

The organization of this paper is as follows: The second section, formulates the CSLHS controllability. Section three recalls some background about bond graph modelling of hybrid systems with ideal switches. In section four the structural controllability of these systems is discussed using bond graph approach and using algebraic characterization.

Graphical conditions and procedures are proposed. Finally, a simple example illustrates the previous results is proposed.

II. CONTROLLABILITY OF CONTROLLED SWITCHING LINEAR HYBRID SYSTEMS

Consider a CSLHS [8], given by equation (1):

( ) ( ( )) ( ( ))

x t A t xB t u (1) Where xRⁿ is the state variable, uR^m is the input variable, :R Q { ,_i i{1, , }} q is a piecewise constant switching function and (_i,x) the hybrid state.

According to values of (t), there exists q configurations,



q



i  

  ₁, . So, A(_i)R^{n n} and B(_i)R^{n m} . The characteristics of CSLHS are:

▪ The dynamical subsystem within each mode has a linear time invariant form,

▪ The admissible region of operation within each mode is the whole state and input space,

Assumptions

1) We suppose that A(_i) and B(_i) matrices are constant on [ ,t t_{0 0}), where ττ_min 0, and constant τ_min is an arbitrarily small and independent of mode i. For instance,

Structural Analysis of Controlled Switching Linear Hybrid Systems Modelled by Bond Graph

Hicham Hihi

Electrical Engineering Department, LGECOS, Cadi Ayyad University, ENSA, Av Abdelkrim khattabi BP, 57540000, Marrakech, Morocco

Volume 3, Issue 3, September 2013 suppose that the dynamics in (1) are given by

u B x A

x (_i)  (_i) over the finite time interval [ ,t t_{k k1}). At time t_k1 the dynamic in interval [t_k1,t_k2) is given by

u B x A

x (_j)  (_j) .

2) We assume that the state vector x(t) does not jump discontinuously at t_k1.

Under these assumptions, the CSLHS controllability of (1) was defined:

Definition 1 [8] Given any pair of hybrid states, (0,x₀) and )

(q,xq , if there exists a timed mode-switching set

1 1

{(_i, ,t_i _i)}^q_i and a corresponding piecewise continuous-finite input signal u(t), such that system (1) evolving under these two distinct inputs is reachable from

)

(0,x₀ to (q,xq) within a finite time interval, then the considered system (1) is controllable, otherwise, system (1) is uncontrollable.

A. Necessary and sufficient algebraic condition

Firstly, we assume that one switching-mode set is known as {_il}^k_l1, where i_l i_l1 for l1,,k1.

Let us define the ( n ,mn^k) matrix

2 1

1 1

2 1

1 , , {0, , 1}

Π( , , ) [ ^k ]

k k

j j j

k

k i i i i j j n

i  i  A A A B _{   } .

Based on the definition of Œ we construct a new matrix Æ as follows:

0 1

Æ ( ) Œ ( )i  i W_i

,…, 1, , 1, , {1, , }

Æ ( ) [Œ ( ,^k i ^k 1 i i i_k)]_{i ik} _q

    



With i₁i, , i_k i_k1.

The joint controllability matrices can be defined as:

0 0 0

[Æ (1) Æ ( )]

W   q ,…,W^k [Æ (1) Æ ( )]^k  ^k q (2)

Wk is the k^th-order joint controllability matrix of the system (1). There exists a joint controllability coefficient k_r ^{of the} system, defined in [8]:

arg min( ( ^l) ( ^l1))

r

l

k  rank W rank W^

Theorem 1 [12] System (1) is controllable, if and only if ( ^kr)

rank W n.

This theorem can be interpreted using geometric approach.

Let us firstly recall some concepts.

Definition 2 (Invariant subspace) [15] Given a matrix A and a subspace BIm B( ), the invariant subspace A B is defined by:

1 1

1

def n

i n

i

A A^ A A^







   

B B B B  B (3)

For system (1), [15] defined a subspace sequence as follows:

1 1 q

i i

i



A







^{B , 1}

1 q

j i j

i



 A









^{j1, 2,}

and

1 k k



^









(4)

The following proposition shows the relationship between the previously defined subspaces and the joint controllability matrix.

Proposition 1 [13] The subspace



(equation 4) and the kth-order joint controllability matrix W^kare linked by the following relation: Im^W^k



.

Based on this proposition, a geometric necessary and sufficient condition is introduced.

Proposition 2 System (1) is controllable, if and only if Rn

  .

Proof. Easy by using theorem 1 and proposition 1.

B. Controllability subspace basis

In this subsection, we give a procedure proven in [13] to calculate



.

Denote the nested subspaces as W₀B₁  B_q ,

1 1

1

1, 2,

q

j j k j

k

A j

 



 





W W W  , and

0 j

j









W W . We

have W₀W₁W₂W and



W. Note that if

1

j j

W W for some j , then W_k W for _j k j and further W_jW

 

. This fact together with dimWn imply that

n n_0 

 

W W , where n₀dimW₀.

Denote min{ :k W_k } n n₀ and

dim 1, ,

k k

n  W k  .

A basis of



can be constructed according to the following procedure:

Procedure 1 1) Choose a group of base vectors

1, , _s1

  in B₁, 2) Expand them to

1 1 2

1, , _s, _s 1, , _s

   _  which form a basis of B₁B₂,

3) Repeat this operation, and write a basis

1, , _n0

  of W₀, Because W₁W₀Im{A_j_k, j1,, ,q k1,,n₀}

1 0 0

Im{ , ,_n ,A_j_k, j 1, , ,q k 1, ,n}

     

4) Write a basis

1, , _n1

  of W₁ by searching the set

1 0 0

{ , ,_n,A_j_k, j1,, ,q k1,,n} from left to right,

Volume 3, Issue 3, September 2013 5) Repeat the operation, and write a basis

0 1

1, , _n, , _nl 1, , _nl}

   _  for W_l. Because

1 Im{ , 1, , , 1 1, , }

l_  l Ajk j q knl_  nl

W W  

=Im

{

₁,..., ,A_{j k}, j 1,...,q,k n_{l 1} 1,...,n

}

nl    _ 





6) By searching the set

1 , 1

{ , , , 1, , , 1, , }

nl Aj k j q k nl nl

      _   and write a

basis

0 1 1

1, , _n, , _nl 1, , _nl, _nl 1, , _nl

   _   _  _ for W_l1. 7) Write

0 1

1 1

Im{ , ,_n, ,_n , ,_n}



   _  (5)

Remark. From the above analysis; a basis for



is of the form





1,1 2,1 1,1 1,1 1,1

1,2 2,2 1,2 2,2 1,2

0 1,0 0 2,0 1,0 0 0,0 1,0 0

1 1 1 1

2 2 1 2

, , , , , ,

, , , , , ,

, , , ,

r

r

n n n rn n n

i i i i i

i i i i i

n i n i i n i i n

b A b A A b A A b

b A b A A b A A b

b A b A A b A A b

  

  

 

(6)

Where

, 0

0 0, 1 , , 1, ,

k k l k n

b W r  i q l r , k1,,n₀. Because the number of vectors in (6) is not more than n ; there are at most n different subsystems whose parameters appear in (6). That is to say; for controllability issues; we may assume qn without loss of generality.

III. BOND GRAPH APPROACH

The bond graph structure junction contains informations on the type of the elements constituting the system, and how they are interconnected, whatever the numerical values of parameters. The structure junction of a switching bond graph can be represented by figure 1. Five fields model the components behavior, 4 that belong to the standard bond graph formalism; - source field which produces energy, - R field which dissipates it, - I and C field which can store it, and the Sw field that is added for switching components.

Fig 1. Structure junction Assumptions:

1) To take into account the absence of discontinuities, we suppose that is no elements in derivative causality in the bond graph model in integral causality, before and after switching.

It can be obtained by assuming that switches commutated by pairs.

2) A switch is considered as a discrete control;

Using the ideal approach, a switch can be modelled as shown in figure 2:

Fig 2. Representation of ideal switch

The corresponding junction matrix is given by equation 7 [2]:

11 13 14 15

0 13 33 34 35

0 14 34 44 45

51 53 54 55

i T

i

T T

i in T

S S S S z

x

S S S S D

D

S S S S T

S S S S

y u

 

 

 

 

 

 

 

 

 

 

 

 

  

 

   

 

 

   

   



(7)

0

Di LD , L is a positive matrix. Let assume that

1

( 33 )

H L IS L ^ is an invertible positive matrix. Then the

second row leads to _{13 34 35}

i T

i in

D  HS FxHS T HS u The third line of (7) gives:

0 ( 14 34 13) ( 44 34 34) ( 45 34 35)

i i

T T T T T

T   S S HS Fx S S HS Tin S S HS u

The substitution in the first line of (7) gives:

11 13 13 14 13 34 15 13 35

( ^T) ( )_ini ( )

x S S HS Fx S S HS T  S S HS u Then, we have:

i ci di ini

xA xB uB T (8) Where A_i(S₁₁S₁₃HS₁₃^T)F , B_ciS₁₅S H S_{13 35} and

14 13 34

Bdi S S H S .

After the commutation, the new inputs and outputs of the junction structure associated with switches become T_in( ) and T₀( ) , which can be related with

ini

T and

oi

T . ( ) is a square diagonal matrix whose diagonal elements are the components of  in the new mode.

0

( ) ( ( )) ( )

( ) ( ) ( ( ))

i i

i i

in in o

in o

T I T T

T T I T

  

  

    

     

 (9)

Using (9) and (7) we have:

0

1

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ' ( ) ' ( ) ( )

c d in

d c d in

o c d in

x A x B u B T

T C x D u D T

D C x D u D T

   

    

   

  

  

  







(10)

11 13 13 14 13 34 2 5 14 34 13

( ) [( ^T) ( ) ( ^{T T T})]

A  S S H S  S S H S k k S S H S F

14 13 34 1 14 13 34 2 5 34 34 1 44 1 3

( ) [( ) ( ) ( ^T )]

Bd  S S H S k S S H S k k S HS k S k k

15 13 35 14 13 34 2 5 34 35 45

( ) [( ) ( ) ( ^T )]

Bc  S S H S  S S H S k k S HS S

5 14 34 13

( ) [ ( ^{T T T})]

Cd   k S S HS , D_c( ) k S HS₅( ₃₄^T ₃₅S₄₅),

5 34 34 1 44 1 3

( ) [ ( ^T )]

Dd   k S HS k S k k ,

1

1( ) [ 13^T 34 2 5( 14^T 34^T 13^T) ] C  L H^ S S k k S S H S F ,

1

34 1 34 2 3 34 34 1 44 1 3

' ( )_d [ ( ^T )]

D  L H S k^ S k k S HS k S k k ,

1

35 34 2 3 34 34 45

' ( )_c [ ( ^T )]

D  L H S^ S k k S HS S ,

1 1( ) [ ( 2 ( )) ( )]

k     I I  ^ ,_{k3( )}    _{(I 2 ( ))} ^1_{( )} ,

1

2( ) [ ( 2 ( )) ( ( )]

k     I I  ^ I  ,

1

4( ) ( 2 ( )) ( ( )) k    I  ^ I  , and

1 5( ) [( 34^T 34 2( ) 44 2( ) 4( )]

k   S HS k  S k  k  ^ .

V u Do

Di xi

zi

To Tin Continuous part

of the system (PC)

Discrete part of the system (PD)

I N T E R F A C E

SDH

ij ec ij

ep Source

field

Intégral R Causality

Junction Structure (0,1,MTF,MGY)

Switch field (Sw)

Controlled event (Users, control)

Event not controlled (Disturbance, ...)

Spontaneous transition

0 ) (xi sij

Sw O : Se: 0 : 0 Sf

0 e

F : ^e0

O

0 f  0

f  F

Volume 3, Issue 3, September 2013 Therefore, for N switches, we have q modes, and

1 1 1 1 0 1

1

[ , )

[ , )

q

c d in

q cq dq in q q

x A x B u B T t t t

x A x B u B T t t t

    



    





 



(11)

IV. STRUCTURAL CONTROLLABILITY

The bond graph concept is an alternate representation of physical systems. Some recent works permit to highlight structural properties of these systems [7,5]. In [7], the structural controllability property is studied using simple causal manipulations on the bond graph model. It is shown that the structural rank concept is somewhat different for bond graph models because it is more precise than for other representations. Our objective is to extend these properties to CSLHS systems.

In the following we note that:

-BG: acausal (without causality) bond graph model

-BGI: bond graph model when the preferential integral causality is affected

-BGD: bond graph model when the preferential derivative causality is affected

-tⁱ: the number of elements in integral causality in BGD_i. i indicate the mode i.

-tⁱ_s: the number of elements in integral causality in BGD_i, when a dualization of the maximum number of continuous input sources is applied (in order to eliminate elements in integral causalities).

- s ij

tSw : the number of elements remaining in integral causality in BGD_i, when a dualization of the maximum number of continuous input sources is applied (in order to eliminate elements in integral causalities) and a dualization of the maximum number of discrete input sources is applied (in order to eliminate these integral causalities).

Let us recall the structural controllability of LTI systems (case q=1).

Theorem 2 [7] The system _i(A B_{i i}) is structurally state controllable if :

- On the BGIi, all dynamical elements in integral causality are causally connected with a continuous control.

- BG-rank[A_i B_i]n. Property 1 [7]

BG-rank[A_iB_i]=rank(S S S_{11 13 15})=nt_sⁱ.

In the next step structural controllability of CSLHS modelled by bond graph is studied. For that, formal representation of structural controllability subspace, denoted as R₀, is given for BG model. It is calculated using causal manipulations. The base of this subspace is used to propose a procedure to study the structural controllability.

A. Graphical necessary and sufficient condition

On the BGD_i (and dualization of inputs sources) there exists

i

ts elements remaining in integral causality and (n t _sⁱ) elements in derivative causality.

i

ts algebraic equations can be written (equation 12):

i ik i 0

k r r

r

g 



 g  (12) -gⁱ_k is either an effort variable e_r for I-element in integral causality or a flow variable f_r for C-element in integral causality,

-gⁱ_r is either an effort variable e_r for I-element in derivative causality or a flow variable f_r for C- element in derivative causality,

-_r^ik is the gain of the causal path between the k^th I or C -elements in integral causality and the r^th I or C-elements in derivative causality.

Let us consider the tⁱ_s row vectors z_kⁱ (k1, , ) t_sⁱ whose components are the coefficients of the variables g_kⁱ and gⁱ_r in equation (12).

Property 2 [6] The t_sⁱ row vectors z_kⁱ (k1, , ) tⁱ_s are orthogonal to the structural controllability subspace vectors of the i^th mode. We write

( ) 1, ,i s i

i k k t

Z  z _

 and R₀^i Im(Z_i).

0

Ri^: uncontrollable subspace in mode i , used to check orthogonality.

Procedure 2: Calculation of R₀^i

1) On the BGD_i, dualize the maximum number of input sources in order to eliminate the elements remaining in integral causality,

2) For each element in integral causality, write the algebraic relation with elements in derivative causality (equation 12), 3) Write a row vector z_kⁱ for each algebraic relation with the causal path gains. (equation 12),

In order to calculate a R₀ⁱ basis, it is enough to find (n t _sⁱ) independent column vectors w r^ir( 1,,n t _sⁱ) . These vectors are gathered in the matrix

( ) 1, , i s

i ir

r n t

W  w _{ }

 .

In the same manner, from the BGD_i (and dualization of inputs sources) (n t _sⁱ) algebraic relations can be calculated (13).

i ir i 0

r k k

k

g 



 g  (13) -gⁱ_r is either an flow variable f_r for I-element in derivative causality or a effort variable e_r for C- element in derivative causality,

-gⁱ_k is either an flow variable f_r for I-element in

integral causality or a effort variable e_r for C- element in integral causality,

Volume 3, Issue 3, September 2013 -_k^ir is the gain of the causal path between the r element in ^th

derivative causality and the k^th element in integral causality.

Suppose now the (n t _sⁱ) column vectors w^ir whose components are the coefficients of the variables g_rⁱ and gⁱ_k in equation (13).

Procedure 3 : Calculation of R₀ⁱ 1) On the BGD_i, dualize the maximum number of

continuous control in order to eliminate the elements in integral causality.

2) For each element remaining in derivative causality, write the algebraic relation with elements in integral causality, (equation 13),

3) Write a column vector w^ir for each algebraic relation with the causal path gains, (equation 13),

with R₀ⁱ Im(Wⁱ).

From the BGD_i (and dualization of inputs sources), the following relation can be calculated for each switch:

' ' 0

i

i ir i

o r k k

k

T g 



 g  (14) -Toi is the variable on the switch, outgoing of the junction structure,

-g'ⁱ_r and g'ⁱ_k can be effort or flow, -_k^ir is defined in equation 13.

From (14) we propose the invariants for the BGD:

Proposition 3 For the hybrid system (1), the invariant associated to each switch for BGDi is given by the inequality constraints relating to the i^th mode:

( )

d

Inv i : ' ' 0

i

i ir i

o r k k

k

T g 



 g  (15) At instant of commutation, from equation 14 and after the annulation of

oi

T , N conditions can be given:

'ⁱ_{r k}^ir 'ⁱ_k 0

k

g 



 g  (16) 'ⁱ_r

g , g'ⁱ_k and _k^ir are defined in equation 13.

Suppose now the ^{( 1)}

s

i i

tSw^{ } column vectors ⁽i¹⁾ i( 1, ..., )

j N

Sw js

w ^{  }

whose components are the coefficients of the variables g'ⁱ_r and g'ⁱ_k in equation (16).

Procedure 4: Calculation of R₀

1) After dualization of the maximum number of input sources in BGD_i, write the relation between each switch element and the dynamical elements,

2) Deduce the ^{( 1)}

s

i i

tSw^{ } invariants for the corresponding BGD_i,

3) Write the conditions of commutation using equation 16, 4) Write a column vector ^{(i1) i}

Sw js

w ^{ } ( j1,,N ) for each algebraic relation with the causal path gains,

5) Check if z w_kⁱ. _{Sw js}^{(i 1) i}0 , and write

11 1 2 1

0 Im( ^{k q q})

R  W W ^ W ^{ } , With

1 ( 1)

1, , , 1, ,

[ ⁱ ]

js ik

i i i i

Sw ki n tisi q

W^{ } W w ^{ }

  

   .

1k1

W : The basis of controllability subspace of initial mode.

Remark. If the sequence of commutation is not ordered, then

1 ( 1) ( )

1, , , , 1, , and

[ ⁱ ]

js js

ik

i i i i i r i

Sw Sw ki n tisi r q i r

W^{ } W w ^{ } w ^{ }

   

   

Proposition 4 System (1) is structurally controllable, if and only if rank W( ^1k1 W^12W^{q 1 q})n.

V. EXAMPLE

Let us consider the following acausal BG model (figure 3):

Fig 3. The acausal BG

We have two complementary switches, then we have two possible configurations: mode 1 (Sw₁ :closed, Sw₂ open) and mode 2 (Sw₁:open, Sw₂:closed) .

The bond graph models in integral causality of mode 1 and 2 are shown in figure 4:

Fig 4. The BGI1 and BGI2 There are four state variables,

Ii

P on I_i (i1,,3), q_c on C . Figure 5 presents the bond graphs in derivative causality after the dualization of inputs sources :

Fig 5. a) BGD1+dualization of sources (mode 1), b) BGD2+dualization of sources (mode 2)

I2 C

0 1

0 1

1 0 I3

I1

Sw1

1

BGD2

1 2 3

4 5

6 7

8 9

10 11

12 13 14

15 16

Df1

0

Sw2 MSe

I2 C

0 1

0 1

1 0 I3

I1

Sw1

1 2 3 1 4 5

6 7

8 9

10 11

12 13 14

15 16

Df1

0

Sw2 MSe

BGD1

I2 C

0 1

MSe

0 1

1 0 I3

I1

Sw1

1

BGI2

1 2 3 4 5

6 7

8 9 10 11

12 13 14

15 16 Df1

0

Sw2 I2

C

0 1

MSe

0 1

1 0 I3

I1

Sw1

1

BGI1

1 2 3 4 5

6 7

8 9 10 11

12 13 14

15 16 Df1

0

Sw2

I2 C

0 1

MSe

0 1

1 0 I3

I1

Sw1

1 2 3 1 4 5

6 7

8 9 10 11

12 13 14

15 16 Df1

0

Sw2

Volume 3, Issue 3, September 2013

■ Calculation of Wⁱ (application of procedure 3) ▪ Calculation of W¹ (mode 1)

The element I₂ is in integral causality, we can write

1 2 3 0

I I I

e e e  , thus z₁¹ (1 11 0).

The algebraic equations corresponding to I₁ and I₃ are given by:

1 2 0

I I

f f  ,

3 2 0

I I

f f  . Then

11 (11 0 0)^t

w  , w¹² (0 11 0)^t. The dynamical element C is not causally connected with I₂, we can write e_c 0. The corresponding vector is w¹³(0 0 0 1)^t . and

1 11 12 13

0 Im{ , , }

R  w w w .

▪ Calculation of W² (mode 2)

The element I₃ is in integral causality and not causally connected with I₂, we can write

3 0

eI  , thus z₁²(0 0 1 0). The element I₂ is in integral causality, we have

1 2 0

I I

e e  , thus z₂² (1 1 0 0).

The algebraic equation corresponding to the element I₁ is given by:

1 2 0

I I

f f  , then w²¹(11 0 0)^t. The dynamical elementC is not causally connected with I₂ and I₃, we can write e_c0, The corresponding vector is w²²(0 0 0 1)^t and R₀²Im{w²¹,w²²}.

■ Inequality constraints (application of proposition 3) The invariants (Inv^d(₁) and Inv^d(₂)) can be computed according to equation (15):

Mode 1:

1 1 2 0

Sw I I

f  f f  ,

2 3 0

Sw I

e e  , mode 2:

1 1 0

Sw I

e e  ,

2 3 1 0

Sw I I

f  f f 

■ Calculation of W^{i 1 i} (Application of procedure 4, steps 1, 2, 3, 4)

We suppose that mode 1 is the initial mode, therefore it is characterized by its controllable subspace

) , , Im( ^{11 12 13}

1

0 w w w

R  and its inequality constraints

1 1 2 0, 2 3 0

Sw I I Sw I

f f f  e e  .

After commutation, we have: -z₁²(0 0 1 0), z₂² (1 1 0 0),

21 (11 0 0)^t

w  , w²² (0 0 0 1)^t and

1 2

1 (110 0)^t

wSw s^  , because ^{2 1 2} . _{Sw s}2 0 z w^  , thus

1 2

1 2 21 22 1 2

0 1

1 0 1 1 0 1 Im( , , ) Im , ,

0 0 0 0 1 0

Sw s w

R w w w



 

     

     

     

                 



and

1 2

( ) 3

rank W^  .

[2→1]:

2 1

2 1 11 12 13 2 1

0 2

1 0 0 1

1 1 0 0

Im( , , , ) Im , , ,

0 1 0 1

0 0 1 0

Sw s w

R w w w w



 

       

       

       

                     



■ Calculation of W(Application of procedure 4, step 5)

11 1 2 2 1 4

0 Im( ^k )

R  W W^W ^ R , so the system is structurally controllable.

VI. CONCLUSION

The structurally controllability of CSLHS systems was presented using simple causal manipulations on the BG. Thus, formal calculation enables us to know the reachable variables;

its checking is immediate on the BGI. On the other hand the BGD enables us to characterize graphically the structural controllability subspaces relating to each mode. A necessary and sufficient condition was given by exploiting these various bases.

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AUTHOR BIOGRAPHY

Hicham Hihi Doctor in Automatic and industrial Computing from Ecole Centrale of Lille, French in 2008. Assistant Professor in the Ecole Nationale des Sciences Appliques of Marrakech, Cadi Ayyad University Morocco, since 2009. His research interests include: analysis of switched and hybrid systems, control theory, state observer and digital signal processing. E-mail:

[email protected]